Fabrication of Gasar Porous Mg Ag_x (x = 0.5, 17.57) Alloys Via the Metal-H Eutectic Directional Solidification Process

Abstract

Gasar porous Mg-Ag alloys have been the focus of research due to their better biodegradable and catalytic performances. Pore structure inside the porous alloy mainly determines the practical aforementioned applications. To gain knowledge on the effect mechanism of alloying elements on pore structure, ordered porous MgAg_x alloys were fabricated through Gasar method. The influence of Ag content on pore structures was analyzed. On one hand, a wider mushy zone with the Ag addition of 0.5 at.% promotes the forming of cellular crystals and then disturbs the continuous growth of pores. On the other hand, compared with the concentrated primary phase, the lamellar distribution of eutectic microstructure is beneficial to obtain regular pores. Based on the binary thermodynamic model, a ternary alloy system model describing the relationship between pore structures and experimental parameters was developed, and the predicted results were in good agreement with the experimental values. This research aimed to provide the theoretical guidance and technical basis for the subsequent detailed control of the porous structure of Gasar magnesium alloys.

Appendices

Appendix 1

The main thermo-physical parameters that directly affect the directional solidification process of an alloy include density ρ_f, latent heat of solidification L, specific heat capacity C_p, thermal conductivity λ_f, convection heat transfer coefficient between cooling water and crystallizer h₀, etc.

• (1) Density

  The density of Mg-Ag alloy can be expressed as follows,

  $$ \rho = \frac{M}{V} $$

  (5)
  $$ V = {{V_{{\text{Ag,m}}} } \mathord{\left/ {\vphantom {{V_{{\text{Ag,m}}} } {\xi_{{{\text{Ag}}}} }}} \right. \kern-0pt} {\xi_{{{\text{Ag}}}} }} $$

  (6)
  $$ V_{{\text{Ag,m}}}^{2/3} = V_{{{\text{Ag}}}}^{2/3} \left[ {1 + \mu_{{{\text{Ag}}}} x_{{{\text{Ag}}}} \left( {\Phi_{{{\text{Ag}}}} - \Phi_{{{\text{Mg}}}} } \right)} \right] $$

  (7)

  where V_Ag,m is the molar volume of the Ag in the Mg-Ag alloy, m³/mol. ξ_Ag is the molar volume fraction of the component Ag. μ_Ag is the constant, which relates to elemental valence, μ_Ag = 0.14 (monovalent alkali metals), μ_Mg = 0.10 (and divalent alkali metals). The parameters for calculation are listed in Table 

  Table III Parameters for calculating density of Mg-Ag alloy²⁸

  III.

  The results obtained are listed in Table 

  Table IV Density of Mg-Ag alloy

  IV.

• (2) Specific heat capacity of Mg-Ag alloys

  Specific heat capacity of Mg-Ag alloys can be calculated through Neumann-Kopp law²⁹

  $$ C_{{\text{P}}} (T) = \sum {C_{{{\text{P}},i}} (T)} x_{i} $$

  (8)

  where C_p,i(T) is the constant pressure specific heat capacity of each element in the alloy, and x_i is the mass percentage of element i in alloy. As for Mg-Ag alloy, C_p, (Mg) = 890 + 4.58 × 10⁻¹ T, C_p, (Ag) = 230 + 1.5 × 10⁻¹ T. The calculated results are listed in Table

  Table V Specific heat capacity of Mg-Ag alloys

   V.

• (3) Thermal conductivity

  The equipment used in the thermal diffusion coefficient test is the laser thermal conductivity analyzer (NETZSCH, German). The thermal diffusion coefficient specimen is first cut by wire-cutting method into Φ12.7 mm × 3 mm. Before the test, the specimens were polished with 1200 mesh, 3000 mesh, and 7000 mesh sandpaper until there were no scratches and then physically polished to ensure that the top and bottom planes of the Mg-Ag alloy specimens were smooth.

  When the temperature of the sample in the container is constant, the thermal diffusion coefficient α of the sample at that temperature can be given by the following equation,²⁹

  $$ \alpha = {{0.1388\,d^{2} } \mathord{\left/ {\vphantom {{0.1388\,d^{2} } {t_{1/2} }}} \right. \kern-0pt} {t_{1/2} }} $$

  (9)

  where d is the thickness of the measured sample and t_1/2 is the half heating time,

  $$ \lambda { = }\alpha \rho C_{{\text{p}}} $$

  (10)

  The results are listed in Table 

  Table VI Thermal conductivity of Mg-Ag

  VI.

• (4) Melt viscosity

  According to the η_F(Pa s) model proposed by Terzieff,³⁰ the viscosity of Mg-Ag alloy can be expressed as,

  $$ \eta_{{\text{F}}} = \sum\limits_{i} {x_{{{\text{Ag}}}} \,\eta_{{{\text{Ag}}}} \left( {1 + \eta^{{\text{E}}} } \right)} $$

  (11)
  $$ \begin{aligned} \eta^{{\text{E}}} = & \alpha \frac{{x_{{{\text{Ag}}}} x_{{{\text{Mg}}}} \left( {\sigma_{{{\text{Ag}}}} - \sigma_{{{\text{Mg}}}} } \right)}}{{x_{{{\text{Ag}}}} \sigma_{{{\text{Ag}}}}^{2} - x_{{{\text{Mg}}}} \sigma _{{{\text{Mg}}}}^{2} }}^{2} + \beta \left\{ {\left[ {1 + \frac{{x_{{{\text{Ag}}}} x_{{{\text{Mg}}}} \left( {M_{{{\text{Ag}}}}^{1/2} - M_{{{\text{Mg}}}}^{1/2} } \right)^{2} }}{{x_{{{\text{Ag}}}} M_{{{\text{Ag}}}}^{1/2} + x_{{{\text{Mg}}}} M_{{{\text{Mg}}}}^{1/2} }}} \right]^{1/2} - 1} \right\} \\ & + 2\gamma \frac{{\Delta H_{{\text{m}}} }}{\Delta T} + \delta \frac{{2x_{{{\text{Ag}}}} x_{{{\text{Mg}}}} |V_{{{\text{Ag}}}} - V_{{{\text{Mg}}}} |}}{{V_{{{\text{Ag}}}} + V_{{{\text{Mg}}}} }} \\ \end{aligned} $$

  (12)

  η is the viscosity of alloy component i, σ₁ and σ₂ are the closest atomic spacings of the component, M₁ and M₂ are the molar masses of the component, and V₁ and V₂ are the molar volumes of the component. α, β, γ and δ are constants, and their values can be obtained by experimental regression, α = 4.05, β = 12.0, γ = 0.115, δ = 2.67 (Table 

  Table VII Parameters for calculating viscosity of Mg-Ag

  VII).

  The results are listed in Table 

  Table VIII Melt viscosity of Mg-Ag alloy

  VIII.

• (5) Latent heat of solidification

  According to the energy conservation law, the released heat L(J/kg) from the transformation of Mg-Ag alloy from liquid to solid can be expressed as

  $$ L = \frac{{H_{{\text{m}}}^{{{\text{liq}}}} (T_{{\text{L}}} ) - H_{{\text{m}}}^{{{\text{hcp}}}} (T_{{\text{L}}} )}}{M} $$

  (13)

  the molar mass of the Mg-Ag alloy (M) can be obtained using linear superposition Mg-Ag, $H_{{\text{m}}}^{{{\text{liq}}}} (T_{{\text{L}}} )$ is the molar enthalpy of the liquid phase at the melting point (J/mol), and $H_{{\text{m}}}^{{{\text{hcp}}}} (T_{{\text{L}}} )$ represents the molar enthalpy of hcp phase at the melting point.

  $$ H_{{\text{F}}} = x_{{{\text{Ag}}}} H_{{{\text{Ag}}}} + x_{{{\text{Mg}}}} H_{{{\text{Mg}}}} + \Delta H_{{\text{m}}} $$

  (14)

  where $x_{{{\text{Ag}}}}$ and $x_{{{\text{Mg}}}}$ are the molar fractions of Ag and Mg in Mg-Ag alloys. The melar enthalpy of Ag and Mg under different states are represented by $H_{{{\text{Ag}}}}$ and $H_{{{\text{Mg}}}}$; $\Delta H_{{\text{m}}}$ is the molar heat of mixing in the alloy.³¹

  $$ \begin{aligned} & \Delta H_{{\text{m}}} = \frac{{x_{{{\text{Ag}}}} x_{{{\text{Mg}}}} V_{{{\text{Ag}},{\text{m}}}}^{2/3} V_{{{\text{Mg}},{\text{m}}}}^{2/3} }}{{\left( {x_{{{\text{Ag}}}} V_{{{\text{Ag}},{\text{m}}}}^{2/3} + x_{{{\text{Mg}}}} V_{{{\text{Mg}},{\text{m}}}}^{2/3} } \right)^{2} }}\frac{{2\left( {x_{{{\text{Ag}}}} V_{{{\text{Ag}},{\text{m}}}}^{2/3} + x_{{{\text{Mg}}}} V_{{{\text{Mg}},{\text{m}}}}^{2/3} } \right)}}{{\left( {n_{{{\text{ws}}}}^{1/3} } \right)_{{{\text{Ag}}}} + \left( {n_{{{\text{ws}}}}^{1/3} } \right)_{{{\text{Mg}}}} }} \\ & \quad p\left[ {\frac{q}{p}\left( {\Delta n_{{{\text{ws}}}}^{1/3} } \right)^{2} - (\Delta \Phi )^{2} - \frac{r}{p}} \right] \\ \end{aligned} $$

  (15)

  where x_Ag and x_Mg are the molar fraction of Ag and Mg, and V_Ag,m and V_Mg,m are the molar volume fractions of Ag and Mg in Mg-Ag alloy. The valency of the metal element, the molar volume of the pure group element V_Ag and the electronegativity all affect the molar volume of Ag in the Mg-Ag alloy, and the interrelationship can be expressed as follows,

  $$ V_{{{\text{Ag}}}}^{2/3} = V_{{{\text{Ag}}}}^{2/3} \left[ {1 + \mu_{{{\text{Ag}}}} x_{{{\text{Mg}}}} \left( {\Phi_{{{\text{Ag}}}} - \Phi_{{{\text{Mg}}}} } \right)} \right] $$

  (16)

  The latent heat of solidification of Mg-Ag alloy can be obtained according to the above equation. The values of latent heat of solidification parameters of Mg-Ag alloy are shown in Tables 

  Table IX Values of latent heat of solidification parameters of Mg-Ag alloy

  IX and

  Table X Calculated results of laten heat of solidification in Mg-Ag alloy

  X.

• (6) Heat transfer coefficient

  The crystallizer-sample interface heat transfer coefficient h_FM is not a constant value during solidification but is related to the interface temperature, solidification rate, sample thermal conductivity, crystallizer thermal conductivity and many other factors.³² Martorano et al. investigated the interfacial heat transfer coefficient of Cu-Sn alloys during directional solidification on a Cu crystallizer, and the average value was about 2000–3000 (W/m²/K), a value similar to the experimentally determined value.^33,34 During the finite element simulation, the interfacial heat transfer coefficient h_FM between the Mg-Ag alloy specimen and the copper crystallizer is taken as approximately 3000 (W/m²/K). Convection heat transfer between crystallizer and cooling water h₀ can be expressed as

  $$ h_{{\text{o}}} = 0.664\lambda \,{\text{pr}}^{1/3} \left( {\frac{{v_{0} }}{vR\sqrt \pi }} \right) $$

  (17)

  where λ(W/m/K) is the thermal conductivity of water, Pr is the Prandtl number of cooling water, and v(m²/s) is the kinematic viscosity of water. The above parameters are obtained at a temperature of T_0, and the values of the parameters are shown in Table 

  Table XI Parameters for calculating h₀

  XI. According to the above equation, h₀ is 4000 (W/m²/K).

Appendix 2

The self-diffusion coefficient of the solute in the melt of a ternary alloy (Mg-Ag-H) is approximately related to the viscosity,³⁵

$$ D\mu d{\text{/kT = 1/2}}\pi $$

(18)

k is the Boltzmann constant, μ is the alloy viscosity coefficient, and d is the average diameter of the Mg, Ag and H atoms. The viscosity factor can be expressed as

$$ \mu = \frac{5}{{16d^{2} }}\left( {\frac{{{\text{km}}_{{{\text{Ag}},{\text{Mg}},{\text{H}}}} T}}{\pi }} \right)^{1/2} $$

(19)

where m_Ag,Mg,H is the reduced mass of Ag, Mg and H atoms together. The relationship among the average atom diameter d, mass of each atom and m_Ag,Mg,H is

$$ d = \left( {\frac{{d_{{{\text{Ag}}}} + d_{{{\text{Mg}}}} + d_{{\text{H}}} }}{3}} \right) $$

(20)

$$ m_{{{\text{Ag}},{\text{Mg}},{\text{H}}}} = \frac{{m_{{{\text{Ag}}}} m_{{{\text{Mg}}}} + m_{{{\text{Ag}}}} m_{{\text{H}}} + m_{{\text{H}}} m_{{{\text{Mg}}}} }}{{m_{{{\text{Ag}}}} + m_{{{\text{Mg}}}} + m_{{\text{H}}} }} $$

(21)

Variation of Mg, Ag and H atomic diameters with temperature

$$ d = 1.288\left( {\frac{m}{{\rho_{{\text{m}}} }}} \right)^{{1/3}} \left[ {{1}{ - }{0}{\text{.112(}}T{/}T_{{\text{m}}} {)}^{{1/2}} } \right] $$

(22)

where ρ_m is the atom density at melt point, and T_m is the melting temperature of atoms. The relative parameters are listed in Table 

Table XII Relative parameters of Mg and Ag

XII.

Appendix 3

Determination of parameters in Tables I and II.

The solubility of hydrogen in the alloy melt $x_{{\text{H}}}$ satisfies Sieverts’ law.

$$ x_{{\text{H}}} = \sqrt {{{p_{{{\text{H}}_{{2}} }} } \mathord{\left/ {\vphantom {{p_{{{\text{H}}_{{2}} }} } {p_{0} }}} \right. \kern-0pt} {p_{0} }}} \exp \left( {{{ - \Delta G_{{\text{m}}}^{0} } \mathord{\left/ {\vphantom {{ - \Delta G_{{\text{m}}}^{0} } {{\text{RT}}}}} \right. \kern-0pt} {{\text{RT}}}}} \right) $$

(23)

where $\Delta G_{{\text{m}}}^{0}$ is the molar free energy change during the dissolution of hydrogen in the molten alloys. For an alloy melt that conforms to the properties of an ideal solution, the $\Delta G_{{\text{m}}}^{0}$ (hydrogen in the alloy melt) can be expressed as

$$ \Delta G_{{\text{m}}}^{0} = \sum\limits_{i} {x_{i} \Delta G_{i}^{0} } $$

(24)

where $x_{i}$ is the molar fraction of alloying element i in the alloy melt, and $\Delta G_{i}^{0}$ is the molar free energy change for the dissolution of hydrogen in alloying element i. For alloy melts that are not ideal solutions, the modified solubility equation is as follows

$$ \Delta G_{{\text{m}}}^{0} = \sum\limits_{i} {x_{i} \Delta G_{i}^{0} } - \Delta G_{{\text{m}}}^{{{\text{ex}}}} $$

(25)

As hydrogen has difficulty forming compounds with the alloying elements, Eq. 25 can be used to calculate the solubility of hydrogen in the alloy melt. Substituting Eq. 25 into Eq. 23 gives the formula for the solubility of hydrogen in the alloy melt.

$$ \ln x_{{\text{H}}} = \sum\limits_{i} {x_{i} \ln x_{{{\text{H}},i}} } + {{\Delta G_{{\text{m}}}^{{{\text{ex}}}} } \mathord{\left/ {\vphantom {{\Delta G_{{\text{m}}}^{{{\text{ex}}}} } {{\text{RT}}}}} \right. \kern-0pt} {{\text{RT}}}} = \sum\limits_{i} {x_{i} \ln \gamma_{i} x_{{{\text{H}},i}} } $$

(26)

where $x_{{{\text{H}},i}}$ is the hydrogen solubility in alloy elements i, and $\gamma_{i}$ is the activity co-efficient of the element i,

$$ \left\{ {\begin{array}{*{20}l} {\Delta G_{{\text{m}}}^{{{\text{ex}}}} = {\text{RT}}\sum\limits_{i} {x_{i} \ln \,\gamma_{i} } } \hfill \\ {{\text{RT}}\,\ln \gamma_{i} = \Delta G_{{\text{m}}}^{{{\text{ex}}}} + \sum\limits_{j = 2}^{n} {\left( {\delta_{ij} - x_{i} } \right)\frac{{\partial \Delta G_{{\text{m}}}^{{{\text{ex}}}} }}{{\partial x_{i} }}} } \hfill \\ \end{array} } \right. $$

(27)

where δ_ij is Kronecker mark (while i ≠ j, δ_ij = 0 and while i = j, δ_ij = 1).

The molar excess free energy of binary alloy $\Delta G_{{\text{m}}}^{{{\text{ex}}}}$ is

$$ \Delta G_{{\text{m}}}^{{{\text{ex}}}} = G_{{\text{m}}} - G_{i} - G_{j} = {\text{RT}}\left[ {\sum {x_{i} \ln \left( {\frac{{\zeta_{i} }}{{x_{i} }}} \right)} + \frac{{\Delta \varepsilon_{{\text{p}}} }}{2kT}} \right] $$

(28)

The molar excess Gibbs free energy of the poly-alloy melt $\Delta G_{{\text{m}}}^{{{\text{ex}}}}$ and the activity coefficient γ_i for component i are

$$ G_{{\text{m}}}^{{{\text{ex}}}} = {\text{RT}}\left[ {\sum\limits_{i = 1}^{n} {x_{i} \ln \frac{{V_{i} }}{{\sum\nolimits_{j = 1}^{n} {x_{j} V_{j} B_{ij} } }}} - \frac{1}{2}\sum\limits_{i = 1}^{n} {Z_{i} x_{i} \left( {\frac{{\sum\nolimits_{j = 1}^{n} {x_{j} B_{ij} \,\ln \,B_{ij} } }}{{\sum\nolimits_{k = 1}^{n} {x_{k} B_{ij} } }}} \right)} } \right] $$

(29)

$$ \begin{aligned} \ln \,\gamma_{i} = & 1 + \ln \frac{{V_{i} }}{{\mathop \Sigma \nolimits_{j = 1}^{n} x_{j} V_{j} B_{ij} }} - \sum\limits_{k = 1}^{n} {\frac{{x_{k} V_{i} B_{ki} }}{{\mathop \Sigma \nolimits_{j = 1}^{n} x_{j} V_{j} B_{kj} }}} \\ & - \frac{1}{2}\left[ {\frac{{Z_{i} \mathop \Sigma \nolimits_{j = 1}^{n} x_{j} B_{ij} \ln B_{ij} }}{{\mathop \Sigma \nolimits_{l = 1}^{n} x_{l} B_{il} }} + \sum\limits_{j = 1}^{n} {\frac{{Z_{j} x_{j} B_{ji} }}{{\mathop \Sigma \nolimits_{l = 1}^{n} x_{l} B_{jl} }}} } \right. \\ & \times \left. {\left( {\ln B_{ji} - \frac{{\mathop \Sigma \nolimits_{t = 1}^{n} x_{t} B_{jt} \ln B_{jt} }}{{\mathop \Sigma \nolimits_{l = 1}^{n} x_{l} B_{jl} }}} \right)} \right] \\ \end{aligned} $$

(30)

In Eqs. 29 and 30, the molar fraction x_i of alloy element i is available, and the molar volume V_i is also available from experimental data. The parameters that need to be determined are the coordination number Z_i and interaction parameter B_ij. Z_i could be calculated through the following formula³⁶

$$ Z_{i} = \frac{{32\sqrt {2\pi } }}{3}\left( {\frac{{\overline{r}_{i}^{3} - r_{i}^{3} }}{{\overline{r}_{i} - r_{i} }}} \right)\frac{{N_{{\text{A}}} }}{{V_{i} }}r_{i} \exp \left( {\frac{{\Delta H_{{{\text{m,}}i}} \left( {T - T_{{{\text{m,}}i}} } \right)}}{{12\,{\text{RTT}}_{{{\text{m,}}i}} }}} \right) $$

(31)

where Avogadro’s constant N_A = 6.022 × 10²³ (mol⁻¹), T_m,i is the melting point of I, ΔH_m,i (J/mol) is the solidification latent heat, $r_{i}$ is the Goldschmidt radius of the metal atom of alloy element i, and $\overline{r}_{i}$ is half the average spacing of the atoms of the alloy element i. This can be approximated as 0.918 times the atomic covalent radius r_conv.

The interaction parameter B_ij can be obtained. Let x_i and x_j in Eq. 29 tend to 0 respectively to derive the infinite dilution activity coefficients ($\gamma_{i}^{\infty }$ and $\gamma_{j}^{\infty }$) for binary alloy melts i–j,

$$ \left\{ {\begin{array}{*{20}l} {\ln \gamma_{i}^{\infty } = 1 - \ln \left( {\frac{{V_{j} B_{ij} }}{{V_{i} }}} \right) - \frac{{V_{i} B_{ji} }}{{V_{j} }} - \frac{1}{2}\left( {Z_{i} \ln B_{ij} + Z_{j} B_{ji} \ln B_{ji} } \right)} \hfill \\ {\ln \gamma_{j}^{\infty } = 1 - \ln \left( {\frac{{V_{i} B_{ji} }}{{V_{j} }}} \right) - \frac{{V_{j} B_{ij} }}{{V_{i} }} - \frac{1}{2}\left( {Z_{j} \ln B_{ji} + Z_{i} B_{ij} \ln B_{ij} } \right)} \hfill \\ \end{array} } \right. $$

(32)

The infinite dilution activity coefficients ($\gamma_{i}^{\infty }$ and $\gamma_{j}^{\infty }$) for binary alloy melts i–j could be determined through experiments; however, data are scarce and measurements are difficult. Thus, the Miedema model in binary alloys can be considered.^36,37,38

According to the Miedema model, binary alloys formed by metal elements i and j could be regarded as the mechanical mixture; the Wigner-Seitz cell of the metal remains unchanged in volume but changes in shape. The heat of mixing of binary alloys $\Delta H_{{\text{m}}}$ is caused by changes in the cell boundary conditions. In the Miedema model, the contribution of three factors to the mixing heat of the alloy is considered. (1) The positive contribution $q\Delta n_{{{\text{ws}}}}^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}}}$ of electron density difference $\Delta n_{{{\text{ws}}}}$ at Wigner-Seitz cell boundaries; q is an empirical constant. (2) The negative contribution $- p(\Delta \phi )^{2}$ of electronegativity differences $\Delta \phi$ between metallic elements; p is an empirical constant. (3) The negative contribution of electron orbital hybridization, expressed as an empirical constant r. On this basis, the heat of mixing of the binary alloy $\Delta H_{{\text{m}}}$ can be expressed as

$$ \Delta H_{{\text{m}}} = f\left( {x_{i} ,V} \right)g\left( {x_{i} ,n_{{{\text{ws}}}} } \right)p\left[ {\frac{q}{p}\left( {\Delta n_{{{\text{ws}}}}^{{{{1} \mathord{\left/ {\vphantom {{1} 3}} \right. \kern-0pt} 3}}} } \right)^{2} - \left( {\Delta \phi } \right)^{2} - \frac{r}{p}} \right] $$

(33)

In Eq. 33,

$$ f\left( {x,V} \right) = \frac{{x_{i} x_{j} V_{{i,{\text{m}}}}^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} V_{{j,{\text{m}}}}^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} }}{{\left( {x_{i} V_{{i,{\text{m}}}}^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} + x_{j} V_{{j,{\text{m}}}}^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} } \right)^{2} }} $$

(34a)

$$ g\left( {x,n_{{{\text{ws}}}} } \right) = 2\frac{{x_{i} V_{{i,{\text{m}}}}^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} + x_{j} V_{{j,{\text{m}}}}^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} }}{{\left( {n_{{{\text{ws}}}}^{{{{1} \mathord{\left/ {\vphantom {{1} 3}} \right. \kern-0pt} 3}}} } \right)_{i} + \left( {n_{{{\text{ws}}}}^{{{{1} \mathord{\left/ {\vphantom {{1} 3}} \right. \kern-0pt} 3}}} } \right)_{j} }} $$

(34b)

where $V_{{i,{\text{m}}}}$ and $V_{{j,{\text{m}}}}$ are the molar volumes of elements i and j in the alloy, respectively (m³/mol), which are related to the chemical valence of the metal element, the molar volume V_i in the pure state and the electronegativity,

$$ V_{{i,{\text{m}}}}^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} = V_{i}^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} \left[ {1 + \mu_{i} x_{j} \left( {\phi_{i} - \phi_{j} } \right)} \right] $$

(35)

where μ_i is a constant related to elemental valence, μ_i = 0.14 (monovalent alkali metals), 0.10 (divalent alkaline earth metals) (Mg), 0.07 (trivalent main group metals and precious metals) and0.04 (other metals). q/p is 9.4, and the value of p is12.3.³⁷ Other parameters such as ϕ, $n_{{{\text{ws}}}}^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}}}$, V^2/3 and r/p^* used in the Miedema model are also determined according to Ref. 37.

The molar excess entropy of a binary alloy $\Delta S_{{\text{m}}}^{{{\text{ex}}}}$ can be expressed as a function of the heat of mixing using the approximation proposed by Tanaka^39,40

$$ \left\{ {\begin{array}{*{20}l} {\Delta S_{{\text{m}}}^{{{\text{ex}}}} = \frac{1}{14.0}\Delta H_{{\text{m}}} \cdot \left( {1/T_{{{\text{m}},i}} + 1/T_{{{\text{m}},j}} } \right)} \hfill & {{\text{Liquid}}} \hfill \\ {\Delta S_{{\text{m}}}^{{{\text{ex}}}} = \frac{1}{15.1}\Delta H_{{\text{m}}} \cdot \left( {1/T_{{{\text{m}},i}} + 1/T_{{{\text{m}},j}} } \right)} \hfill & {{\text{Solid}}\, {\text{solution}}} \hfill \\ \end{array} } \right. $$

(36)

Substituting Eqs. 33 and 36 into Eq. 25, the activity coefficient of the binary alloy elements (γ_i and γ_j) are calculated.

$$ \left\{ {\begin{array}{*{20}c} {{\text{RT}}\,\ln \gamma_{i} = \alpha_{ij} f_{ij} \left\{ {{A \mathord{\left/ {\vphantom {A B}} \right. \kern-0pt} B} + x_{j} {{\left[ {\left( {E \cdot D + F \cdot C} \right) \cdot B - G \cdot A} \right]} \mathord{\left/ {\vphantom {{\left[ {\left( {E \cdot D + F \cdot C} \right) \cdot B - G \cdot A} \right]} {B^{2} }}} \right. \kern-0pt} {B^{2} }}} \right\}} \\ {{\text{RT}}\,\ln \gamma_{j} = \alpha_{ij} f_{ij} \left\{ {{A \mathord{\left/ {\vphantom {A B}} \right. \kern-0pt} B} + x_{i} {{\left[ {\left( {E \cdot D + F \cdot C} \right) \cdot B + G \cdot A} \right]} \mathord{\left/ {\vphantom {{\left[ {\left( {E \cdot D + F \cdot C} \right) \cdot B + G \cdot A} \right]} {B^{2} }}} \right. \kern-0pt} {B^{2} }}} \right\}} \\ \end{array} } \right. $$

(37)

In Eq. 37

$$ A = x_{i} x_{j} \left[ {1 + \mu_{i} x_{j} \left( {\phi_{i} - \phi_{j} } \right)} \right]\left[ {1 + \mu_{j} x_{i} \left( {\phi_{j} - \phi_{i} } \right)} \right] $$

(38a)

$$ B = x_{i} V_{i}^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} \left[ {1 + \mu_{i} x_{j} \left( {\phi_{i} - \phi_{j} } \right)} \right] + x_{j} V_{j}^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} \left[ {1 + \mu_{j} x_{i} \left( {\phi_{j} - \phi_{i} } \right)} \right] $$

(38b)

$$ C = x_{i} \left[ {1 + \mu_{i} x_{j} \left( {\phi_{i} - \phi_{j} } \right)} \right] $$

(38c)

$$ D = x_{j} \left[ {1 + \mu_{j} x_{i} \left( {\phi_{j} - \phi_{i} } \right)} \right] $$

(38d)

$$ E = 1 + \mu_{i} \left( {\phi_{i} - \phi_{j} } \right)\left( {x_{j} - x_{i} } \right) $$

(38e)

$$ F = 1 + \mu_{j} \left( {\phi_{j} - \phi_{i} } \right)\left( {x_{i} - x_{j} } \right) $$

(38f)

$$ \begin{aligned} G = & V_{i}^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} \left[ {1 + \mu_{i} x_{j} \left( {\phi_{i} - \phi_{j} } \right)} \right] - x_{i} V_{i}^{2/3} \mu_{i} \left( {\phi_{i} - \phi_{j} } \right) \\ & - V_{j}^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} \left[ {1 + \mu_{j} x_{i} \left( {\phi_{j} - \phi_{i} } \right)} \right] + x_{j} V_{j}^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} \mu_{j} \left( {\phi_{j} - \phi_{i} } \right) \\ \end{aligned} $$

(38g)

$$ \left\{ {\begin{array}{*{20}l} {\alpha_{ij} = 1 - \frac{1}{14.0}T\frac{{T_{{{\text{m,}}i}} + T_{{{\text{m,}}j}} }}{{T_{{{\text{m,}}i}} T_{{{\text{m,}}j}} }}} \hfill & {{\text{Liquid}}} \hfill \\ {\alpha_{ij} = 1 - \frac{1}{15.1}T\frac{{T_{{{\text{m,}}i}} + T_{{{\text{m,}}j}} }}{{T_{{{\text{m,}}i}} T_{{{\text{m,}}j}} }}} \hfill & {{\text{Solid}} \,{\text{solution}}} \hfill \\ \end{array} } \right. $$

(38h)

In addition, the solubility of hydrogen in pure Mg could be obtained.

$$ {\text{ln}}\,S_{{\text{H}}} = { 6}.{558} - {2533}/T + {1}/{2}\,{\text{ln }}\left( {p_{{{\text{H2}}}} /p_{0} } \right) $$

(39)

After calculation, the required physical parameters for the model are listed in Tables I and II.